Chapter 10.7, Problem 16P

a.

To determine

To show: The $\varphi \text{ and }\psi$ must satisfy $\varphi \left(x\right)+\psi \left(x\right)=f\left(x\right),\text{ and }-{\varphi }^{\prime }\left(x\right)+{\psi }^{\prime }\left(x\right)=0$.

b.

To determine

To show: The $u\left(x,t\right)=\frac{1}{2}\left(f\left(x-at\right)\right)+f\left(x+at\right)$ by solving the equation obtained in part a for $\varphi \text{ and }\psi$.

c.

To determine

To show: The equation $f\left(x-at\right)=\{\begin{array}{l}2,-1+at<x<1+at\\ 0,\text{ otherwise}\end{array}$ and also determine $f\left(x+at\right)$.

d.

To determine

To sketch: The graph of the equation $u\left(x,t\right)=\frac{1}{2}\left(f\left(x-at\right)+f\left(x+at\right)\right)$ for $t=0,t=\frac{1}{2a},t=\frac{1}{a},\text{ and }t=\frac{2}{a}$.